Abstract
We consider the problem of a government that wishes to promote replacing old cars with new ones via a vehicle scrappage program. Since these programs increase consumer’s willingness to pay for a new car, manufacturers (or dealers) could respond strategically by raising their prices. In a two-period game between a government and a manufacturer, we find equilibrium prices and subsidy levels. Our results demonstrate that price levels are increasing over time and are higher than in the benchmark case where no subsidy is offered. Furthermore, if consumers act strategically, then the equilibrium price levels will be higher than in the scenario where they behave myopically.
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Acknowledgements
We would like to thank two anonymous reviewers for their helpful comments. Research supported by NSERC, Canada, Grant RGPIN-2021-02462.
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Appendix: Proofs
Appendix: Proofs
1.1 Proof of Proposition 1
We start by solving for the manufacturer to obtain its reaction to the subsidy announced by the government. Recall that the manufacturer’s profit is given by
First, we consider the second-period manufacturer’s profit. Taking the derivative of \(\Pi _{2}\) with respect to \(p_{2}\), we get
Substituting for \(p_{2}\) in \(\Pi\) yields
Differentiating with respect to \(p_{1}\) gives
Substituting in \(p_{2}\), we obtain the manufacturer’s reaction functions, that is,
To solve the government’s problem, we write down the Lagrangian
where \(\mu\) is the Lagrange multiplier appended to the target constraint. Developing the above equation and inserting for \(p_{1}\) and \(p_{2}\) from (15), we get
Assuming an interior solution, the first-order optimality conditions are
Solving, we obtain
The values of \(p_{1}\) and \(p_{2}\) can be found by inserting s into their formulas.
1.2 Proof of Proposition 2
As in Proposition 1, we solve for the manufacturer and get the following reaction function:
To solve the the government’s problem, we introduce the Lagrangian
where \(\mu\) is the Lagrange multiplier. Substituting for p and solving, we obtain the following unique subsidy and value of the Lagrange multiplier:
Substituting in the price, manufacturer’s profit and government’s cost, we obtain the \({\tilde{p}}\), \({\tilde{\Pi }}\) and \({\tilde{C}}\) in the statement of the Proposition. We note that the equilibrium is indeed interior, and the demands are positive:
1.3 Proof of Proposition 4
Same as proposition 1, we solve for the manufacturer and we get the following price functions:
The Lagrangian function of the government is given by:
which results in the following solution:
which leads to the following prices:
We check for nonnegativity of demands:
and
If \(\Gamma \ge \frac{7\theta +6\gamma -6\theta \beta +4\beta \gamma }{ 8\left( \theta +2\gamma \right) }\), then demand \(d_{1}^{\eta -1}\) will be set equal to zero.
Finally,
1.4 Proof of Proposition 5
To solve for a Markov-perfect (feedback) Stackelberg equilibrium, we solve the game backward, that is, we start by the second stage.
1.5 Second-period equilibrium
For any given \(s_{2}\) announced by the government, the manufacturer solves the following optimization problem:
Introduce the manufacturer’s Lagrangian
where \(\lambda _{2}\) is the Lagrange multiplier appended to the constraint \(p_{2}\ge 0\). The first-order optimality conditions are
Solving the first equation, gives
Now, we consider the second-period government’s optimization problem, which is given by
Substituting for \(p_{2}\) in \(d_{2}\), the above optimization problem becomes
Introduce the second-period government’s Lagrangian
where \(\eta _{2}\) is the Lagrange multiplier appended to the constraint \(s_{2}\ge 0\). The first-order optimality conditions are
Solving the first equation, we obtain
Substituting in \(p_{2}\) yields
The second-period demand is given by
To wrap up, in (16) and (17), we express the second-period strategies in terms of the first-period decision variables.
1.6 First-period equilibrium (or overall equilibrium problem)
The manufacturer overall optimization problem is as follows:
where \(p_{2}\left( s_{1},p_{1}\right)\) and \(d_{2}\left( s_{1},p_{1}\right)\) have been determined in the previous step and the product \(p_{2}\left( s_{1},p_{1}\right) d_{2}\left( s_{1},p_{1}\right)\) plays the role of a salvage value in the current optimization problem.
Substituting for the second-period equilibrium strategies, the above optimization problem becomes:
The Lagrangian is given by
where \(\lambda _{1}\) is the Lagrange multiplier appended to the constraint \(p_{1}\ge 0\).
The first-order optimality conditions give
Now, we turn to the government’s optimization problem. Substituting for \(p_{1}\left( s_{1}\right)\) in (16)–(18) and in the demands, we get
Inserting for the above values in the following optimization problem
we get
The Lagrangian is given by
where \(\mu\) and \(\eta _{1}\) are the Lagrange multipliers appended to the target constraint and \(s_{1}\ge 0\), respectively.
The first-order optimality conditions are
From the second condition, we can get \(s_{1}\) as function of the model’s parameters and the Lagrange multipliers, that is,
Inserting for \(s_{1}\) in \(p_{1}\,\)we get
Substituting for \(s_{1}\left( \lambda _{1},\lambda _{2},\eta _{2}\right)\) in \(s_{2}\left( s_{1}\right)\) and \(p_{2}\left( s_{1}\right)\), we obtain
We have four cases to consider:
-
1.
\(s_{2}>0\) and \(p_{2}>0\) \(\Rightarrow\) \(\eta _{2}=\lambda _{2}=0\). Then,
$$\begin{aligned} \eta _{2}= & {}\; \Gamma -1<0, \end{aligned}$$(36)$$\begin{aligned} p_{2}\left( \lambda _{2},\eta _{2}\right)= & {}\; 1-\Gamma >0. \end{aligned}$$(37)Clearly, \(s_{2}\) is negative and therefore a contradiction.
-
2.
\(s_{2}=0\) and \(p_{2}>0\) \(\Rightarrow\) \(\eta _{2}<0\) and \(\lambda _{2}=0\). Then,
$$\begin{aligned} \eta _{2}= & {}\; \frac{\left( 7\theta ^{2}+3\gamma ^{2}+10\theta \gamma \right) \left( \Gamma -1\right) }{11\theta \gamma }<0, \end{aligned}$$(38)$$\begin{aligned} \lambda _{2}= & {}\; \frac{\left( 7\theta ^{2}+3\gamma ^{2}+10\theta \gamma \right) \left( \Gamma -1\right) }{11\theta \gamma }>0. \end{aligned}$$(39) -
3.
\(s_{2}=0\) and \(p_{2}=0\) \(\Rightarrow\) \(\eta _{2}<0\) and \(\lambda _{2}>0\). Then,
$$\begin{aligned} s_{2}= & {}\; \frac{2\left( 7\theta +3\gamma \right) \left( \Gamma -1\right) }{14\theta ^{2}+17\gamma \theta }<0, \end{aligned}$$(40)$$\begin{aligned} \lambda _{2}= & {}\; \frac{\left( \Gamma -1\right) \left( 7\theta ^{2}+3\gamma ^{2}+10\theta \gamma \right) }{\theta \left( 14\theta +17\gamma \right) }<0, \end{aligned}$$(41)The fact that \(\lambda _{2}\) is strictly negative is a contradiction.
-
4.
\(s_{2}>0\) and \(p_{2}=0\) \(\Rightarrow\) \(\eta _{2}=0\) and \(\lambda _{2}>0\). Then, a contradiction.
Consequently, the only admissible solution is
Substituting for these values in (32) and (33), we obtain
Clearly, \(p_{1}\left( \lambda _{1}\right)\) is strictly positive and therefore \(\lambda _{1}\) must be equal to zero. Therefore, the final value of \(p_{1}\) is
and
which is positive for
To determine the Lagrange multiplier associated with the target constraint, it suffices to substitute for the equilibrium values in \(\frac{\partial {\mathcal {L}}_{1}}{\partial s_{1}}=0\) to obtain
We check for the nonnegativity of demands. First, we have
Using the constraint \(d_{1}^{\ge \eta }+d_{1}^{\eta -1}+d_{2}=\Gamma\), we get
So the total demand in the first period is positive. Moreover,
Using \(\Gamma >\frac{2\theta +\gamma }{3\theta +2\gamma }\), we have
Consider now \(d_{1}^{\eta -1}:\)
If \(\Gamma >\frac{\left( 1-\beta \right) \left( 2\theta +\gamma \right) }{ \left( \theta +\gamma \right) }\), then \(d_{1}^{\eta -1}=0\).
1.7 Proof of Proposition 6
The manufacturer profits and subsidy costs in both settings are given by:
Considering the circumstances in propositions 4 and 5, \(\check{C}-{\hat{C}}\) and \(\check{\Pi } -{\hat{\Pi }}\) are increasing in \(\Gamma\) and positive at \(\Gamma =\frac{ 2\theta +\gamma }{3\theta +2\gamma }\), that is to say, \(\check{C}>{\hat{C}}\) and \(\check{\Pi }>{\hat{\Pi }}\).
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Zaman, H., Zaccour, G. Subsidies and pricing strategies in a vehicle scrappage program with strategic consumers. Cent Eur J Oper Res 32, 457–481 (2024). https://doi.org/10.1007/s10100-023-00867-z
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DOI: https://doi.org/10.1007/s10100-023-00867-z